Understanding the On-policy state distribution for episodic tasks with $\gamma \in (0,1)$
This question is really getting at the meaning of the discount factor in Markov decision processes. There are actually two, equivalent ways of interpreting the discount factor.
The first is probably what you're familiar with: the $i^{th}$ step receives a reward of $\gamma^{i}r_i$ instead of just $r_i$.
The second, equivalent interpretation, is that at every step, we have a $1-\gamma$ chance of immediately ending the episode. Assuming we don't end the episode, we receive the full $r_i$ reward. This is what Sutton and Barto mean by termination. Note that under the second interpretation, we have a $\gamma^i$ probability of reaching the $i^{th}$ step, hence the connection to the first interpretation.
Your definition of $\eta_{\gamma}(s)$ looks correct to me. Alternatively, we can write it in Sutton's, Barto's notation like
$$\eta_{\gamma}(s) = h(s) + \sum_{\bar s}\eta(\bar s)\sum_{a}\gamma \pi(a | \bar s) p(s | \bar s, a)$$